(Anglais) We review the intimate connection between (super-)gravity close to a spacelike singularity
(the \BKL-limit") and the theory of Lorentzian Kac{Moody algebras. We show that in this
limit the gravitational theory can be reformulated in terms of billiard motion in a region of
hyperbolic space, revealing that the dynamics is completely determined by a (possibly in nite)
sequence of re
ections, which are elements of a Lorentzian Coxeter group. Such Coxeter
groups are the Weyl groups of in nite-dimensional Kac{Moody algebras, suggesting that these
algebras yield symmetries of gravitational theories. Our presentation is aimed to be a self-
contained and comprehensive treatment of the subject, with all the relevant mathematical
background material introduced and explained in detail. We also review attempts at making
the in nite-dimensional symmetries manifest, through the construction of a geodesic sigma
model based on a Lorentzian Kac{Moody algebra. An explicit example is provided for the
case of the hyperbolic algebra E10, which is conjectured to be an underlying symmetry of
M-theory. Illustrations of this conjecture are also discussed in the context of cosmological
solutions to eleven-dimensional supergravity.